an interval-parameter fuzzy two-stage stochastic program for water resources management under...

European Journal of Operational Research 167 (2005) 208–225

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Stochastics and Statistics

An interval-parameter fuzzy two-stage stochastic programfor water resources management under uncertainty

Imran Maqsood a, Guo H. Huang b,c,*, Julian Scott Yeomans d

a Environmental Systems Engineering Program, Faculty of Engineering, University of Regina, Regina, Sask., Canada S4S 0A2b Sino-Canada Center of Energy and Environmental Research, Hunan University, Changsha, China

c Faculty of Engineering, University of Regina, Regina, Sask., Canada S4S 0A2d Management Science Area, Schulich School of Business, York University, Toronto, Ont., Canada M3J 1P3

Received 12 September 2002; accepted 4 August 2003

Available online 8 June 2004

Abstract

This study presents an interval-parameter fuzzy two-stage stochastic programming (IFTSP) method for the planning

of water-resources-management systems under uncertainty. The model is derived by incorporating the concepts of

interval-parameter and fuzzy programming techniques within a two-stage stochastic optimization framework. The

approach has two major advantages in comparison to other optimization techniques. Firstly, the IFTSP method can

incorporate pre-defined water policies directly into its optimization process and, secondly, it can readily integrate

inherent system uncertainties expressed not only as possibility and probability distributions but also as discrete intervals

directly into its solution procedure. The IFTSP process is applied to an earlier case study of regional water resources

management and it is demonstrated how the method efficiently produces stable solutions together with different risk

levels of violating pre-established allocation criteria. In addition, a variety of decision alternatives are generated under

different combinations of water shortage.

� 2004 Elsevier B.V. All rights reserved.

Keywords: Decision analysis; Environment; Programming; Uncertainty; Water resources

1. Introduction

For many decades, problems involving the

efficient allocation of water have challenged water

* Corresponding author. Address: Environmental Engineer-

ing Program, Faculty of Engineering, University of Regina,

Regina, Sask., Canada S4S 0A2. Tel.: +1-306-585-4095; fax:

+1-306-585-4855.

E-mail address: [email protected] (G.H. Huang).

0377-2217/$ - see front matter � 2004 Elsevier B.V. All rights reserv

doi:10.1016/j.ejor.2003.08.068

resource managers. During this time period, con-

troversial and conflict-laden water allocation is-sues between competing municipal, industrial, and

agricultural interests have intensified (Huang and

Chang, 2003; Wang et al., 2003). Recently, in-

creased population shifts and shrinking water

supplies have exacerbated this user competition.

This competitiveness increases under conditions of

natural variation and as the concerns for water

quantity and quality grow. Serious problems canarise from poorly planned water allocation systems

ed.

mailto:[email protected]

I. Maqsood et al. / European Journal of Operational Research 167 (2005) 208–225 209

when faced with disadvantageous climate andriver-flow conditions. For a long time, increasing

water demand was satisfied by developing new

sources of water. However, the significant eco-

nomic and environmental costs associated with

developing new water sources make this approach

unsustainable and unlimited expansion can no

longer be the primary means.

To address the above concerns, innovativeoptimization techniques have been developed for

allocating and managing water in more efficient

and environmentally benign ways. Numerous

optimization techniques for a diverse range of

system complexities have been proposed (Kulcar,

1996; Psilovikos, 1999; Hof and Bevers, 2000;

Needham et al., 2000; Cai et al., 2001; Barbosa

and Pricilla, 2002; Gang et al., 2003; Luo et al.,2003). However, many system parameters are

highly uncertain and their interrelationships can be

extremely complicated (Babaeyan-Koopaei et al.,

2003; Byun et al., 2003). For example, spatial and

temporal variations in such system components as

stream flows and water-allocation targets can ex-

ist, and measures of net system benefits can con-

tain a number of stochastic factors. Thesecomplexities become further compounded not only

by interactions between the uncertain parameters

but also through additional economic implica-

tions. As a result, the inherent complexity and

stochastic uncertainty existing in real-world water-

resource decision-making have essentially placed

them beyond the conventional deterministic opti-

mization methods.Stochastic, fuzzy, and interval-parameter pro-

gramming techniques have been employed in order

to counteract these difficulties (Huang and Moore,

1993; Huang et al., 1994; Wagner et al., 1994;

Chang et al., 1996; Russell and Campbell, 1996;

Kira et al., 1997; Shih, 1999; Chang and Chen,

2000; Haurie and Moresino, 2000; Guo et al.,

2001). Several studies applying these techniques towater resources management and planning prob-

lems have been undertaken, including: Abrisham-

chi et al. (1991) who studied reservoir planning for

irrigation districts using a chance-constrained

optimization model; Huang (1996) who employed

an interval-parameter model for water quality

management within an agricultural system, and

Jairaj and Vedula (2000) who optimized a multi-reservoir system using fuzzy programming. An-

other optimization technique that has received

attention is two-stage stochastic programming

(TSP) (Anderson, 1968; Kall, 1979; Louveaux,

1980; Birge, 1985; Birge and Louveaux, 1988;

Gassmann, 1990; Eiger and Shamir, 1991; Lustig

et al., 1991; Sen, 1993; Edirisinghe and Ziemba,

1994; Ruszczynski and Swietanowski, 1997; Ber-aldi et al., 2000; Dai et al., 2000; Darby-Dowman

et al., 2000; Yoshitomi et al., 2000; Zhao, 2001).

TSP proves effective for the analysis of medium- to

long-term planning problems in which an exami-

nation of policy scenarios is desired and the system

data is characterized by uncertainty. The funda-

mental idea behind stochastic programming is the

concept of recourse, which refers to the ability totake corrective action after a random event has

taken place. In TSP, an initial decision is made

based on uncertain future events. When these fu-

ture uncertainties are later resolved, a recourse or

corrective action is taken. The initial decision is

called the first-stage decision, and the corrective

action is called the second-stage decision. The

objective function for such a two-stage recourseexample would be to minimize the expected costs

of all applicable decisions taken over the two

periods. TSP methodologies and solution algo-

rithms have been applied to diverse applications.

For example, Wang and Adams (1986) used the

two-stage optimization framework for planning

optimal reservoir operations, where hydrologic

uncertainty and the seasonality of reservoir inflowswere modelled as periodic Markov processes.

Optimal release volumes in successive time periods

were determined such that the expected total re-

wards resulting from the operations were maxi-

mized. Pereira and Pinto (1991) proposed a

stochastic optimization approach for a multi-res-

ervoir hydroelectric system operating under

uncertainty. In their study, a given probability wasassociated with each range of inputs that could

occur over different stages of the planning horizon.

The optimal solution was obtained by applying a

decomposition principle in which each sub-prob-

lem was iteratively solved using linear program-

ming. Ferrero et al. (1998) examined hydrothermal

scheduling of multi-reservoir systems using a

210 I. Maqsood et al. / European Journal of Operational Research 167 (2005) 208–225

two-stage algorithm. More recently, Huang andLoucks (2000) proposed an inexact two-stage sto-

chastic programming (ITSP) model to address the

uncertainties. In their study, the concept of inexact

optimization was incorporated within a two-stage

stochastic programming framework. The model

was applied to a case study of water resources

management. However, the main limitation of the

ITSP model remains in its over-simplification offuzzy membership information into intervals,

resulting in reduced system reliability.

In many real-world applications, results pro-

duced by optimization techniques can be rendered

highly questionable if the modelling inputs cannot

be expressed with precision (Yeomans and Huang,

2003; Yeomans et al., 2003). Consequently, the

alternative optimization approaches of chance-

constraint programming (CPP), fuzzy programming

(FP), and interval-parameter programming (IPP)

have been employed to account for this impreci-

sion. Unfortunately, the CPP and FP methods

cannot be effectively linked to the economic con-

sequences of violating predefined system con-

straints, which is an essential feature for related

policy analyses (Liu et al., 2003). Moreover, whileCPP and FP can effectively express the stochastic

aspects of a model’s right-hand-sides, they cannot

capture independent uncertainties in the parame-

ters of either the left-hand-sides or the cost coef-

ficients. Conversely, IPP proves to be an effective

procedure to deal with uncertainties in a model’s

left-hand-sides, but encounters difficulties when

the right-hand-sides are highly uncertain. Fur-thermore, while in IPP uncertain input parameters

are estimated by discrete intervals, the lower and

upper bounds of these intervals may also be

uncertain; thereby generating a dual uncertainty

into the data.

Therefore, one approach to potentially address

all of these uncertainties would be to integrate

both FP and IPP into the TSP framework. Such anapproach would directly incorporate model

uncertainties expressed as fuzzy membership

functions, probability density functions and dis-

crete intervals into its solution procedure. More

importantly, it could also be used to quantitatively

analyze a variety of policy scenarios that are

associated with different levels of economic pen-

alties experienced when promised policy targetsare violated. This combination of methods leads to

the idea of creating an interval-parameter fuzzy

two-stage stochastic programming (IFTSP) solu-

tion approach.

In this paper, an IFTSP model extending the

ITSP work of Huang and Loucks (2000) is devel-

oped and applied to a case study of water resource

planning under uncertainty. The goal of thisapplication is to determine an efficient allocation

of water to competing municipal, agricultural and

industrial interests, while simultaneously account-

ing for the inherent system uncertainties that occur

under different planning scenarios. It will be

demonstrated how IFTSP produces solutions with

higher net benefits than ITSP, while simulta-

neously reducing the system risks that are ex-pressed as interval ranges. Furthermore, it will be

shown how IFTSP can be used to help decision-

makers identify and evaluate alternative system

designs and to determine which of these designs

can most efficiently achieve the desired system

objectives.

2. IFTSP modelling formulation

Because of population and economic growth,

municipal, industrial, and agricultural water de-

mands have been increasing. These disparate

groups of water users need to know how much

water they can expect in order to make appropri-

ate decisions to support their various activities andinvestments. This information is necessary for

planning since, if the promised water cannot be

delivered due to insufficient supply, users will have

to either obtain water from higher-priced alterna-

tives or curb their development plans. For exam-

ple, municipal residents may have to curtail the

watering of lawns, industries may have to reduce

production levels or increase water recycling rates,and farmers may not be able to conduct irrigation

as planned. These actions will result in increased

costs or decreased benefits for regional develop-

ment. For instance, farmers who know there is

only a small chance of receiving sufficient water in

a dry season are not likely to make a major

investment in irrigation infrastructure. Similarly,


industries knowing that they will have to limittheir water consumption are not likely to under-

take the development of water intensive projects.

It is thus necessary for the available water be

effectively allocated to minimize any or all associ-

ated penalties or negative consequences. Here, the

associated penalties mean the acquisition of water

from higher-priced alternatives, and the negative

consequences are generated from the curbing ofthe development plans (Howe et al., 2003; Wang

and Du, 2003).

The water resources allocation problem can be

formulated as one for maximizing the expected

value of net system benefits. Based on the local

water management policies, a target quantity of

water can be allocated for each user group. If this

quantity is subsequently delivered, it will result innet benefits; otherwise, penalties will be incurred. In

this problem, a decision of water allocation target

needs to be made at the beginning facing future

uncertainties of river flow; at a future time, when

the uncertainties of water flow are quantified, a

recourse action can then be taken. Thus, decision of

water allocation made at the beginning is called the

first-stage decision, and the recourse decision iscalled the second-stage decision. This leads to a TSP

problem. Huang and Loucks (2000) introduced an

interval-parameter TSP formulation (i.e. ITSP) to

tackle both the inherent parameter uncertainty and

the difficulty in approximating the uncertainty with

appropriate probability distributions.

2.1. Definitions for interval-parameter

programming

Prior to formulating our IFTSP model for wa-

ter resource planning, we first introduce and re-

view several ancillary definitions used in earlier

interval-parameter or grey system studies (Huang

et al., 1995) that will be implemented through a

series of model transformations to assist withcomputational efforts.

Definition 1. Let x denote a closed and bounded

set of real numbers. A grey number x� with a

known upper and lower bound but with unknown

distribution information is defined as an interval

for x such that

x� ¼ ½x�; xþ� ¼ ft 2 x j x� 6 t6 xþg; ð1Þ

where x� and xþ represent the lower and upper

bounds of x�, respectively. When x� ¼ xþ, x� be-

comes a deterministic number, i.e. x� ¼ x� ¼ xþ.

Definition 2. For x�, the following relationships

hold:

x� P 0 iff x� P 0 and xþ P 0; ð2Þ

x� 6 0 iff x� 6 0 and xþ 6 0: ð3Þ

Definition 3. For x� and y�, their order relations

are as follows:

x� 6 y� iff x� 6 y� and xþ 6 yþ; ð4Þ

x� < y� iff x� < y� and xþ < yþ: ð5Þ

Definition 4. The whitened value of x� is defined as

a deterministic number with its value lying be-tween the upper and lower bounds of x�:

x� 6 x�v 6 xþ; ð6Þwhere x�v represents the whitened value of x�.

Definition 5. For x�, Signðx�Þ is defined as follows:

Signðx�Þ ¼ 1 if x� P 0;�1 if x� < 0:

�ð7Þ

Definition 6. For x�, its absolute value jxj� is de-

fined as follows:

jxj� ¼ x� if x� P 0;�x� if x� < 0:

�ð8Þ

Thus

jxj� ¼ x� if x� P 0;�xþ if x� < 0;

�ð9Þ

and

jxjþ ¼ xþ if x� P 0;�x� if x� < 0:

�ð10Þ

Definition 7. A grey (interval-parameter) system is

defined as a system containing information pre-

sented as grey numbers (i.e. interval).


Definition 8. A grey decision is defined as a deci-

sion made within a grey system.

Definition 9. Let R� denote a set of grey numbers.

A grey vector X� is a tuple of grey numbers, and a

grey matrix X� is a matrix whose elements are grey

numbers:

X� ¼ fx�i ¼ ½x�i ; xþi � j 8ig; X� 2 fR�g1�n; ð11Þ

X� ¼ fx�ij ¼ ½x�ij ; xþij � j 8i; jg; X� 2 fR�gm�n:

ð12Þ

Definition 10. For grey vectors and matrices:

X� P 0 iff x�ij P 0; 8i; j;X� 2 fR�gm�n

; mP 1; ð13Þ

X�6 0 iff x�ij 6 0; 8i; j;

X� 2 fR�gm�n; mP 1: ð14Þ

Definition 11. Let � 2 fþ;�;�;�g be a binary

operation on grey numbers. For x� and y�:

x� � y� ¼ ½minfx � yg;maxfx � yg�;x� 6 x6 xþ; y� 6 y6 yþ: ð15Þ

In case of division, it is assumed that y� do not

contain a zero. Hence, we have

x� þ y� ¼ ½x� þ y�; xþ þ yþ�; ð16Þ

x� � y� ¼ ½x� � yþ; xþ � y��; ð17Þ

x� � y� ¼ ½minfx� yg;maxfx� yg�;x� 6 x6 xþ; y� 6 y6 yþ; ð18Þ

x� � y� ¼ ½minfx� yg;maxfx� yg�;x� 6 x6 xþ; y� 6 y6 yþ: ð19Þ

Definition 12. Let R� denote a set of grey num-

bers. A grey linear programming (GLP) model can

be defined as follows:

max f � ¼ C�X� ð20aÞs:t:

A�X�6B�; ð20bÞ

X� P 0; ð20cÞ

x�j ¼ grey decision variable; x�j 2 X�; ð20dÞ

where

A� 2 fR�gm�n; B� 2 fR�gm�1

;

C� 2 fR�g1�nand X� 2 fR�gn�1

:

Under given interval information for parameters

A�, B� and C�, the GLP model provides opti-

mized grey solutions for the decision variables,

x�j opt, 8j, and the objective function value, f �opt, as

follows:

x�j opt ¼ ½x�j opt; xþj opt�; xþj opt P x�j opt; 8j; ð21Þ

f �opt ¼ ½f �

opt; fþopt�; f þ

opt P f �opt: ð22Þ

Remark 1. When the model elements contain high

levels of uncertainty, model (20) may generate grey

solutions with high grey degrees (i.e. solutions in

which uncertainties are expressed as large greyintervals). Obviously, the higher the grey degree of

the solutions, the lower the effectiveness and use-

fulness of those results. When grey solutions have

very high grey degrees, they may be of limited

practical use for decision makers.

One potential approach for decreasing solution

uncertainties, and thus increasing system effective-ness, is to more carefully consider the stipulated

uncertain characteristics. In this regard, a grey

fuzzy linear programming (GFLP) model can be

used to effectively communicate membership

information for admissible violations of the system

objective and constraints into its optimization

framework. Therefore, a GFLP model can be for-

mulated wherein solutions with lower grey degreesand improved applicability would be expected.

Definition 13. A GFLP submodel can be formu-

lated as follows:

max k�; ð23aÞ

C�X�6 f � þ ½1� k��½f þ � f ��; ð23bÞ

A�X�6B� þ ½1� k��½Bþ � B��; ð23cÞ


X� P 0; ð23dÞ

x�j ¼ grey decision variable; x�j 2 X�; ð23eÞ

06 k� 6 1; ð23fÞwhere f þ is the most desirable system objective

value; f � is the least desirable system objectivevalue; k� is the control decision variable corre-

sponding to the degree (membership grade) to

which X� solution fulfils the fuzzy objective or

constraints.

Remark 2. Based on the principle of fuzzy flexible

programming, the k� value corresponds to the

membership grade of satisfaction of a fuzzy deci-sion. Specifically, the flexibility in the constraints

and fuzziness in the system objective, which are

represented by fuzzy sets and denoted as ‘‘fuzzy

constraints’’ and a ‘‘fuzzy goal’’, respectively, are

expressed by the membership grade (k�) corre-

sponding to the degree of satisfaction for the

constraints/objective.

Remark 3. To determine the tolerance interval (i.e.

Df ¼ f þ � f �) for the system objective in the

above GFLP submodel, the GLP submodel (20)

should be solved before solving submodel (23).

Remark 4. The GFLP submodel (23) will have

optimal grey solutions as follows:

x�j opt ¼ ½x�j opt; xþj opt�; 8j; ð24Þ

k�opt ¼ ½k�opt; kþopt�; ð25Þ

f �opt ¼ ½f �

opt; fþopt�: ð26Þ

2.2. An IFTSP formulation for water allocation

modelling

Suppose that a water resource manager is faced

with the problem of determining the most appro-

priate way in which to allocate a scarce water

supply between the competing requirements of a

municipality, an industrial unit, and an agricul-

tural sector. The future availability of this water

supply is uncertain, but has been estimated by themanager to fall within reasonably established va-

lue ranges. The manager is aware that the indus-

trial unit and agricultural sector would curtail

certain planned expansion activities should an

inadequate water supply be made available to

them; the municipal interest holds the largest

economic influence within the region. Since

uncertainties exist, the manager needs to create aplan that effectively allocates the uncertain supply

of water to the three users in order to maximize the

overall system benefits while simultaneously con-

sidering the system disruption risks attributable to

the uncertainty. Because the supply uncertainties

have been expressed in terms of intervals, this

water-allocation problem can be expressed as the

following ITSP formulation using the intervalnotation introduced above:

max f � ¼Xmi¼1

B�i W

�i �

Xmi¼1

Xn

j¼1

pjC�i S

�ij ð27aÞ

subject to

Xmi¼1

ðW �i � S�

ij Þ6 q�j ; 8j ð27bÞ

[water availability constraints],

S�ij 6W �

i 6W �i max; 8i ð27cÞ

[allowable water allocation constraints],

S�ij P 0; 8i; j ð27dÞ

[non-negativity and technical constraints],

where f � is the net system benefit ($); B�i is the net

benefit to user i per m3 of water allocated ($/m3),

(first-stage revenue parameter); W �i is the target

water allocation that is promised to user i (m3),(first-stage decision variable); C�

i is the loss to user

i per m3 of water not delivered, Ci > Bi ($/m3),

(second-stage cost parameter); S�ij is the shortage

of water to user i under flow level j, the amount by

which Wi is not met when the seasonal flow is qj(m3), (second-stage decision variable); q�j is the

amount of seasonal flow under flow level j (m3),

(random variable); W �i max is the maximum allow-

able allocation amount for user i (m3); pj is the

probability of occurrence of flow level j; m is the


total number of water users; n is the total number

of flow levels; i is the water user (in our example,

m ¼ 3 and i ¼ 1; 2; 3 with i ¼ 1 for the munici-

pality, i ¼ 2 for the industrial user, and i ¼ 3 for

the agricultural sector); j is the flow level in our

example, n ¼ 3 and j ¼ 1; 2; 3 with j ¼ 1 repre-

senting low flows, j ¼ 2 representing medium

flows, and j ¼ 3 representing high flows.Model (27) forms a two-stage linear program

because the water-allocation targets W �i must be

set at the first stage before the stream flows q�j are

known, while the water-shortages S�ij are deter-

mined during the second stage when the stochastic

stream flows are known but the allocation

amounts have been fixed.

In Eq. (27b), the seasonal flows are uncertainand can only be quantified by fuzzy membership

functions. In fact, for many real-world problems,

quality of available information is generally poor,

and is often presented as either vague values or

discrete intervals. For example, uncertainties in

stream flows may be presented as discrete inter-

vals; at the same time, the lower and upper bounds

of these intervals may also not be known withcertainty. This leads to dual uncertainties for the

flow data, as shown in Fig. 1. Therefore, a new

concept of interval membership function (IMF) is

defined to reflect this complexity of uncertainty.

The uncertainty of IMF is derived from those of its

lower and upper limits. Thus, we have

A�ðq�j Þ ¼ ðq�j Þi; lA

�ððq�j ÞiÞ

� �j i 2 I ; q�j 2 Q�; 8j

n o;

ð28Þ

1

0 ~

±jq+

jqjq~

~

jq~

+jq

jq +— — —jq

A(q±j)µ

Fig. 1. Interval membership function for.

where A�ðq�j Þ is an interval-fuzzy set in Q�,

lA�ððq�j ÞiÞ is the degree of interval-membership

function of element ðq�j Þi in A�ðq�j Þ for each

q�j 2 Q�, and Q� is a collection of flows denoted

by ðq�j Þi, which are water flows expressed as

intervals to user i under flow level j. Let

maxi2I

ðq��j Þi ¼ q�j and mini2Iðq

��j Þi ¼ q�

j; 8j; ð29Þ

where q�j and q�jare the upper and lower bounds,

respectively, of the fuzzy water flows ðq��Þ under

any flow level j. Obviously, each of the upper andlower bounds of this fuzzy q

��j must also be rep-

resented by intervals.

According to Zimmermann (1985), decision-

makers may establish an aspiration level (f þ), and

a tolerable interval (Df ¼ f þ � f �) for the objec-

tive they desire to achieve, and each of the con-

straints can be modelled as a fuzzy set. Thus, to

better communicate fuzzy objective and con-straints, as well as the flow uncertainties, model

(27) can be converted into the following interval-

parameter fuzzy two-stage stochastic program-

ming (IFTSP) problem:

max k; ð30aÞ

Xmi¼1

B�i W

�i �

Xmi¼1

Xn

j¼1

pjC�i S

�ij P f � þ Df k; ð30bÞ

Xmi¼1

ðW �i � S�

ij Þ6 q�j � Dq��j k; 8i; ð30cÞ

W �i 6W þ

i max þ DWi maxk; 8i; ð30dÞ

S�ij 6W �

i ; 8i; j; ð30eÞ

S�ij P 0; 8i; j; ð30fÞ

06 k6 1; ð30gÞwhere k and S�

ij are decision variables. k value

corresponds to the degree of satisfaction of the

fuzzy objective or constraints. k value ranges be-

tween 0 (zero) and 1 (one). The values closer to 1correspond to a solution with a higher possibility

of satisfying the constraints/objective under more

advantageous system conditions; conversely, the

values near 0 relate to a solution that has a low


possibility of satisfying the constraints/objective

under conservative considerations. In model (30),

we have: Df ¼ f þ � f �; DWi max ¼ W þi max � W �

i max,

where W þi max and W �

i max are upper and lower

bounds, respectively, of the maximum allowable

allocation amount for user i; and Dq��j ¼ q�j � q�

j¼

½q�j ; qþj � � ½q�j; qþ

j� ¼ ½q�j � qþ

j; qþj � q�

j�, where q�

jis

lower interval of lower bound seasonal flow, qþj

is upper interval of lower bound seasonal flow, q�jis lower interval of upper bound seasonal flow, and

qþj is upper interval of upper bound seasonal flow

under a flow level j.The model (30) can deal with uncertainties de-

scribed as not only intervals but also probability

distributions and membership functions. When

W �i are known as a deterministic value, the model

can be transformed into two sets of deterministic

submodels, which correspond to the lower and

upper bounds of the desired objective (Huang,

1996). This transformation process is based on a

manual algorithm, which is different from normal

interval analysis and best/worst case analysis

(Huang et al., 1993). The resulting solutions pro-

vide stable intervals for the objective function anddecision variables with different levels of risk in

violating the constraints. They can be easily

interpreted for generating decision alternatives.

According to Huang and Loucks (2000), let

W �i ¼ W �

i þ DWizi, where W �i have a deterministic

value, DWi ¼ W þi � W �

i and zi 2 ½0; 1�. Here zi aredecision variables that are used to determine an

optimized set of target values (W �i ) that are essen-

tial for the related policy analyses. For, when W �i

approach their upper bounds (i.e. when zi ¼ 1), the

system benefit will be the highest as long as the

water demands are well satisfied; however, this is

associated with a higher risk of penalty when the

promised amount is not delivered. Conversely,

when W �i reach their lower bounds (i.e. when

zi ¼ 0), we may have a lower system benefit but, atthe same time, a lower risk of violating the prom-

ised amounts (and thus lower risk of system-failure

penalties). Therefore, it is difficult to determine

whether W þi or W �

i will correspond to the desired

lower bound of system benefit.

If W �i are considered as uncertain inputs, the

existing methods for solving interval LP problems

cannot be used directly (Huang, 1996). Therefore,

an optimized set of target values can be obtainedby including zi in model (30) as decision variables.

Thus, by incorporating values of W �i , Df , DWi max

and Dq�j within the model (30), we can obtain:

max k; ð31aÞ

Xmi¼1

B�i ðW �

i þ DWiziÞ �Xmi¼1

Xn

j¼1

pjC�i S

�ij

� kðf þ � f �ÞP f �; ð31bÞ

Xmi¼1

ðW �i þ DWizi � S�

ij Þ

þ k½q�j � qþj; qþj � q�

j�6 q�j ; 8j; ð31cÞ

W �i þ DWizi þ kðW þ

i max � W �i maxÞ6W þ

i max; 8i;ð31dÞ

�DWizi þ S�ij 6W �

i ; 8i; j; ð31eÞ


06 zi 6 1; 8i; ð31gÞ

06 k6 1; ð31hÞwhere S�

ij , zi and k are decision variables.Model (31) can be solved by converting it into

two deterministic submodels (32) and (33). Since

the objective is to maximize net system benefit,

submodel with k corresponding to f þ (i.e. most

desirable system objective value) is desired first.

The combination of upper bounds for benefit

coefficients and decision variables and the lower

bounds for cost terms would correspond to f þ.According to Huang (1996), the submodel to find

f þ is

max k; ð32aÞ

Xmi¼1

Bþi ðW �

i þ DWiziÞ �Xmi¼1

Xn

j¼1

pjC�i S

�ij


Xmi¼1

ðW �i þ DWizi � S�

ij Þ þ kðq�j � qþjÞ6 qþj ; 8j;

ð32cÞ


W �i þ DWizi þ kðW þ

i max � W �i maxÞ6W þ

i max; 8i;ð32dÞ

�DWizi þ S�ij 6W �

i ; 8i; j; ð32eÞ


06 zi 6 1; 8i; ð32gÞ

06 k6 1; ð32hÞwhere S�

ij , zi and k are decision variables. Let S�ij opt,

zi opt and kopt be the solutions of submodel (32).

The optimized water-allocation can be performed

by calculating W �i opt ¼ W �

i þ DWizi opt, which cor-

responds to the extreme upper bound of system

benefit under uncertain inputs of water-allocation

amounts. According to Huang (1996), the sub-

model corresponding to f � can be formulated as

follows:

max k; ð33aÞ

Xmi¼1

B�i ðW �

i þ DWizi optÞ �Xmi¼1

Xn

j¼1

pjCþi S

þij


Xmi¼1

ðW �i þ DWizi opt � Sþ

ij Þ þ kðqþj � q�jÞ6 q�j ; 8j;

ð33cÞ

�DWizi opt þ Sþij 6W �

i ; 8i; j; ð33dÞ

Sþij P S�

ij opt; 8i; j; ð33eÞ

06 k6 1; ð33fÞwhere Sþ

ij and k are decision variables. Submodels

(32) and (33) and are deterministic LP problems.

Thus, according to Huang et al. (1993), solutions

for model (31) under the optimized water-alloca-

tion are

S�ij opt ¼ ½S�

ij opt; Sþij opt�; 8i; j; ð34Þ

k�opt ¼ ½k�opt; kþopt�; ð35Þ

f �opt ¼ ½f �

opt; fþopt�; ð36Þ

where k�opt, f�opt and S�

ij opt are optimal system reli-

ability, optimal objective function value, and

optimal water shortage to user i under flow level j,respectively. S�

ij opt, k (corresponding to f þ) and f þopt

are from solution of submodel (32), and Sþij opt, k

(corresponding to f �) and f �opt are from submodel

(33). Thus, the optimum water-allocation to the

given users is

A�ij opt ¼ W �

i opt � S�ij opt; 8i; j; ð37Þ

where A�ij opt is the optimal water allocation to user i

under flow level j, which is obtained by subtractingthe optimal water shortages S�

ij opt (the second-stage

decision variables) from the optimized target water

allocation W �i opt (the first-stage decision variables).

The main advantage of advancing the prescribed

two-stage stochastic modelling approach is that

different policies for water resources management

can be quantitatively incorporated within the

modelling framework through the first-stage deci-sion variable, W �

i . If the model were simply con-

structed with the A�ij (instead of W �

i and S�ij ) as the

decision variables, then the related water manage-

ment policies would not have been reflected in the

modelling process and thus the related policy

implications would not have been tackled.

Fig. 2 shows the general framework of the IF-

TSP model. It is based on three optimizationtechniques namely TSP, IPP and FP. Each tech-

nique has a unique contribution in enhancing the

model’s capability in dealing with uncertainties in

the system information and the associated policies.

For example, the probability distributions and

policy implications were handled though TSP, the

uncertainties presented as discrete intervals were

reflected though IPP, and the system imprecisenesswas addressed through FP. The modelling outputs

offered solutions under different scenarios of allo-

cation targets, which are helpful for generating

decision alternatives.

In the following, solution algorithm of the IF-

TSP model with the objective being maximized is

presented in a pseudo-code format as follows:

Step 1. Formulate IFTSP model (30).

Step 2. Reformulate the IFTSPmodel by introduc-

ing W �i ¼ W �

i þ DWizi where DW ¼ W þi �

W �i and zi 2 ½0; 1�, this leads to model (31).

ITSP Model

Two-stage stochastic programming

Policy (target)

Imprecise information

Fuzzy programming

Solutions under different scenarios of allocation targets

IFTSP Model

Generation of decision alternatives

Interval-parameter programming

IFTSP upper bound submodel

IFTSP lower bound submodel

Discrete intervals

Probability distributions

Uncertain information and policy

Fig. 2. Schematic of the IFTSP methodology.


Step 3. Transform model (31) into two submodels,

where the objective is to maximize f �.

Step 4. Formulate f þ submodel (32).Step 5. Solve the f þ submodel, and obtain S�

ij opt

and zi opt.Step 6. Calculate W �

i opt ¼ W �i þ DWizi opt, where

W �i opt have deterministic values.

Step 7. Calculate f þopt.

Step 8. Formulate f � submodel (33).

Step 9. Solve the f � submodel, and obtain Sþij opt.

Step 10. Calculate f �opt.

Step 11. Solutions of the IFTSP model are

S�ij opt ¼ ½S�

ij opt; Sþij opt�; 8i; j;

k�opt ¼ ½k�opt; kþopt�

and

f �opt ¼ ½f �

opt; fþopt�:

Step 12. Thus, we have A�ij opt ¼ W �

i opt � S�ij opt, 8i; j.

Step 13. STOP.

Compared with ITSP (Huang and Loucks,

2000), the IFTSP approach provides more infor-

mation regarding trade offs among system benefits,

certainty and reliability. As the actual value of

each variable or parameter varies within its two

bounds, the system benefit may change corre-spondingly between f �

opt and f þopt with a variety of

reliability levels. In comparison, when the same

problem is solved directly through an ITSP

method (Huang and Loucks, 2000), the uncer-

tainties in model’s right and left hand sides are all

expressed as intervals. The main limitation of the

ITSP is its over-simplification of fuzzy member-

ship information into intervals. This leads to thelack of system reliability information as defined by

k�opt in the obtained solutions.

3. Case study

3.1. Overview of the studied system

Consider a case in which a water manager is

responsible for allocating water in a dry season

from an unregulated reservoir to three users: a

municipality, an industrial unit, and an agricultural

±2W

±3W±

1WReservoir

Municipal AgriculturalIndustrial

Q

Fig. 3. Schematic of water-allocation to multiple users.

Table 1

Allowable water allocations (in 106 m3) and related economic data (in $/m3)

Activity User

Municipal (i ¼ 1) Industrial (i ¼ 2) Agricultural (i ¼ 3)

Maximum allowable allocation (W �i max) 7 7 7

Water allocation target (W �i ) ½1:5; 2:5� ½2:0; 4:0� ½3:5; 6:5�

Net benefit when water demand is satisfied (B�i ) ½90; 110� ½45; 55� ½28; 32�

Reduction of net benefit when demand is not

delivered (C�i )

½220; 280� ½60; 90� ½50; 70�

Table 2

Stream flow distribution (in 106 m3) and the associated probabilities

Activity Flow level

Low (j ¼ 1) Medium (j ¼ 2) High (j ¼ 3)

Seasonal flow rate (q�i ) ½½3:2; 3:8�; ½4:2; 4:8�� ½½7; 9�; ½11; 13�� ½½14; 16�; ½18; 20��Probability (pj) 0.2 0.6 0.2


sector (Fig. 3). The industrial unit and agriculturalsector are expanding and want to know how much

water they can expect. If water supply is insuffi-

cient, they will curtail their expansion plans. Tables

1 and 2 show the related water resources and eco-

nomic data (Huang and Loucks, 2000). If the

promised water is delivered, a net benefit to the

local economy will be generated for each unit of

water allocated. However, if the promised water isnot delivered, either the water must be obtained

from higher priced alternatives or the demand must

be curtailed by reduced production, resulting in a

reduced net system benefit.

Therefore, the problems under consideration

are how to effectively allocate water to the three

users to achieve a maximum benefit under uncer-

tainty while incorporating water policies with theleast risk of system disruption. Since uncertainties

exist in terms of intervals, probability distributions

and fuzzy membership functions, and a link to apredefined policy is desired, the IFTSP is consid-

ered to be a feasible approach for this type of

planning problem.

3.2. Result analysis

Table 3 shows results obtained through the

IFTSP model. It is indicated that solutions for theobjective function value and most of the non-zero

decision variables related to the agricultural and

industrial water uses are intervals, while those re-

lated to municipal water use are deterministic

values. In case of insufficient water, allocation

should firstly be guaranteed to the municipality,

secondly to the industry, and lastly to the agri-

culture. This is because municipal use brings thehighest benefit when water demand is satisfied and

is subject to the highest penalty if the promised

Table 3

Solution of the IFTSP model under optimized water-allocation targets (in 106 m3)

Activity Probability (%) Municipal (i ¼ 1) Industrial (i ¼ 2) Agricultural (i ¼ 3)

Target (W �i opt) 2.5 4.0 4.6

Shortage (S�ij opt) under a flow level of

Low (j ¼ 1) 20 0 ½1:7; 3:3� 4.6

Medium (j ¼ 2) 60 0 0 ½0; 4:1�High (j ¼ 3) 20 0 0 0

Allocation (A�ij opt) under a flow level of

Low (j ¼ 1) 20 2.5 ½0:7; 2:3� 0

Medium (j ¼ 2) 60 2.5 4.0 ½0:5; 4:6�High (j ¼ 3) 20 2.5 4.0 4.6

Net benefit ($106) f �opt ¼ ½325; 577�

System reliability k� ¼ ½0:26; 0:94�


water is not delivered; whereas, the industrial and

agricultural uses correspond to lower benefits and

penalties.

The optimized water flow patterns and the

associated allocation targets are presented in Fig.4. It is shown that the flow allocation patterns vary

among different users with uncertain characteris-

tics. This is resulted from the uncertainties of the

inputting W �i (policies), B�

i (benefits), C�i (costs)

and q�i (stream flows) as well as the complexities of

their interactions.

In Table 3, the solutions of S�11 ¼ 0, S�

21 ¼½1:7; 3:3� � 106 and S�

31 ¼ 4:6� 106 m3 indicatethat, under low streamflow levels, there will be no

Fig. 4. Optimized water-allocation pattern

shortage of water (in reference to the optimized

water-allocation target of 2.5 · 106 m3) for muni-

cipal water use. However, some shortages of

½1:7; 3:3� � 106 and 4.6 · 106 m3 may exist (in ref-

erence to the optimized water-allocation targets of4.0 · 106 and 4.6 · 106 m3) for industrial and agri-

cultural uses, respectively, with the probability of

occurrence being 20%. Similarly, the results of

S�12 ¼ S�

22 ¼ 0 and S�32 ¼ ½0; 4:1� � 106 m3 indicate

that, under medium flows, there will be no short-

ages of water for municipal and industrial uses. The

situation is more ambiguous for agricultural water

use. There may be no water shortage underadvantageous conditions when the other users do

s under low, medium and high flows.


not consume the full amounts of the targeted de-mands and/or the actual q�2 value approaches its

upper level; however, under demanding conditions,

the shortage may become as high as 4.1 · 106 m3

with the probability of 60%. Likewise, the solutions

of S�31 ¼ S�

32 ¼ S�33 ¼ 0 indicate that, under high

streamflow levels, there may be zero shortage of

water, and thus water will be fully allocated to the

three users with a probability of 20%.Table 4 describes solutions under different wa-

ter-allocation scenarios. The solution when all W �i

reach their lower bounds represents a situation

when the manager is conservative regarding the

water availability, and thus promises the lower

bound target values to all users. This leads to a

plan with both lower shortage and lower alloca-

tion (i.e. lower S�ij and A�

ij ), but a higher risk ofwasting available water; thus the net benefit under

this condition is $½269; 365� � 106. Conversely, the

solution when all W �i reach their upper bounds is

applicable when the manager is optimistic

regarding water availability. Thus, a plan with

both higher shortage and higher allocation is

generated, where risks of water insufficiency may

exist but the net benefit is $½185; 618� � 106. At thetargets’ upper bounds, the resulting plan will be

effective in high-runoff years when all targeted

water demands are delivered; but it will become

risky under low-runoff conditions due to the deficit

of water availability and the relevant penalties.

Table 4

Solutions under different scenarios of water-allocation targets (in 106

W �i ¼ W �

i W �i ¼

i ¼ 1 i ¼ 2 i ¼ 3 i ¼ 1

Target (W �i ) 1.5 2.0 3.5 2.5

Shortage (S�ij )

j ¼ 1 0 ½0; 0:3� ½2:2; 3:5� 0

j ¼ 2 0 0 0 0

j ¼ 3 0 0 0 0

Allocation (Aþij )

j ¼ 1 1.5 ½1:7; 2� ½0; 1:3� 2.5

j ¼ 2 1.5 2.0 3.5 2.5

j ¼ 3 1.5 2.0 3.5 2.5

Objective (f �) $½269; 365� � 106

Lambda (k�) ½0:10; 0:99�

The solution when all W �i equal their mid-values

[W ðmidÞi ¼ ðW �

i þ W þi Þ=2, i ¼ 1; 2; 3] corresponds to

a situation when water availability stands between

conservative and optimistic conditions, the net

benefit is $½227; 493� � 106 with a reliability level of

k ¼ ½0:37; 0:99�. Here, k ¼ 0:37 corresponds to a

scenario under advantageous system conditions

(with less concern of satisfying the constraints and

aspiration) with a net benefit of 493 · 106; incomparison, k ¼ 0:99 relates to a more conserva-

tive scenario under disadvantageous system con-

ditions (with more emphasis on satisfying the

constraints and aspiration) with a reduced net

benefit (227 · 106).Table 5 presents four decision alternatives gen-

erated under different combinations of water

shortage at lower and upper levels using a 22 (i.e.,two-level with two-variable) factorial design ap-

proach. The alternatives were produced by adjust-

ing the deficit values and thus allocation values

between upper and lower bounds of non-zero S�ij opt.

The intervals for S�ij opt are useful for decision-

makers to justify the generated alternatives directly,

or to adjust the allocation scheme when they are not

satisfied with the recommended alternatives. De-spite variations in S�

21, alternatives 1 and 2 (where

S�32 ¼ S�

32) will lead to significantly higher system

benefits than alternatives 3 and 4 (where S�32 ¼ Sþ

32).

The effects of S�21, S

�32 and S�

21S�32 combined effect are

)23, )124 and )4, respectively; which means that

m3)

W þi W �

i ¼ W ðmidÞi

i ¼ 2 i ¼ 3 i ¼ 1 i ¼ 2 i ¼ 3

4.0 6.5 2.0 3.0 5.0

½1:7; 3:3� 6.5 0 ½0:2; 1:8� 5.0

0 ½0; 6� 0 0 [0, 3]

0 0 0 0 0

½0:7; 2:3� 0 2.0 ½1:2; 2:8� 0

4.0 ½0:5; 6:5� 2.0 3.0 ½2; 5�4.0 6.5 2.0 3.0 5.0

$½185; 618� � 106 $½227; 493� � 106

½0:38; 0:92� ½0:37; 0:99�

Table 5

Alternatives obtained from the solutions when all W �i reach their upper bounds

Alternative S�21 S�

32 f�opt f ðmidÞ

opt (1) (2) Divisor Effect Identification

1 ) ) ½383; 618� 501 960 1672 4 419 Average

2 + ) ½368; 549� 459 712 )46 2 )23 S�21

3 ) + ½269; 493� 381 )42 )248 2 )124 S�32

4 + + ½185; 477� 331 )50 )8 2 )4 S�21S

�32

Note: f ðmidÞopt ¼ ðf�

opt þ f þoptÞ=2.


S�32 (i.e. water-shortage to agricultural use under

medium flows) has a more significant effect on the

system benefit than S�21 (i.e. water-shortage to

industrial use under low flows) and S�21S

�32 (i.e.

combined shortage to industrial and agricultural

uses under low to medium flows). The negative sign

for S�32 indicates that the system benefit will decrease

as the deficit increases. This result is consistent with

the above analysis of the relationship between S�32

and system benefit. Therefore, effective planning

for water allocation to the agricultural sector at

medium seasonal flow is more important forimproving the system’s performance than that of

the industrial sector. Similar post-optimality anal-

yses can also be conducted for solutions under

other scenarios of water-allocation targets.

In Table 5, economic impacts of variations in

water supply and demand are also determined by

letting the W �i reach their upper bound. Generally,

for each IFTSP solution under a given scenario ofwater-allocation targets, lower shortage values

correspond to more advantageous conditions. For

example, alternative 1 (where S�32 ¼ S�

32 and

S�21 ¼ S�

21) corresponds to a condition when water

shortage values reach their lower bounds, which is

advantageous (with the upper bound system ben-

efit). In comparison, alternative 4 (where S�32 ¼ Sþ

32

and S�21 ¼ Sþ

21) is based on a more demandingcondition under which water shortage values reach

their upper bounds, leading to a lower-bound

system benefit. These alternatives reflect relation-

ships between economic consideration and re-

sources availability.

3.3. Policy analysis

Solutions of the IFTSP model provide desired

water allocation patterns, which maximize both

the system benefit and feasibility. The complexities

associated with the water-allocation amounts arise

mainly from limited supply and increasing de-

mand. Therefore, the observed variations in the

values of W �i could reflect different policies for

water resources management.

An optimistic policy corresponding to the up-

per-bound system benefit may be subject to a high

risk of system-failure penalties; while a too con-

servative policy may lead to a waste of resources.

Solutions under other policy scenarios can also be

obtained by having W �i equal different determinis-

tic values, representing different options for tradingoff among system benefit, reliability, and safety.

4. Advantages of IFTSP over TSP

Model (27) can also be solved through a con-

ventional two-stage programming (TSP) method

by making all interval parameters equal to theirmid-point values. By this way, a net benefits of

fopt ¼ $425� 106 is obtained, which is indeed one

of many alternatives from the IFTSP. Although

further sensitivity analysis can be undertaken, each

TSP solution can only provide an individual re-

sponse to variations of the uncertain inputs.

Therefore, sensitivity analysis does not accurately

reflect interactions among various uncertainties(Huang and Loucks, 2000).

The problem can also be solved directly through

an ITSP method (Huang and Loucks, 2000) by

expressing uncertainties in model’s right and left

hand sides as intervals. The main limitation of the

ITSP is its over-simplification of fuzzy member-

ship information into intervals. This leads to the

lack of system reliability information as defined byk�opt in the obtained solutions. Fig. 5 presents a

comparison of the objective function values ob-

tained through the conventional ITSP and IFTSP

Fig. 5. Comparison of net benefits obtained through ITSP and IFTSP models.


approaches. The ITSP provides a net benefit of

f �opt ¼ $½260; 592� � 106, whereas the IFTSP leadsto a net benefit of $½325; 577� � 106 with the pos-

sibility of satisfying the system constraints and

aspiration level being k� ¼ ½0:26; 0:94�. It is shownthat the IFTSP model leads to a higher mid value

and a smaller interval than the ITSP. The raised

benefit corresponds to a reduced possibility in

satisfying the constraints and aspiration; and the

increased system certainty (i.e. the shrunk intervalwidth) is based on a reduced certainty on the

possibility of satisfying the constraints and aspi-

ration. Thus, the IFTSP approach provides more

information regarding trade offs among system

benefits, certainty and reliability. As the actual

value of each variable or parameter varies within

its two bounds, the system benefit may change

correspondingly between f �opt and f þ

opt with a vari-ety of reliability levels.

The IFTSP approach has an advantage in

providing an effective linkage between the pre-de-

fined water policies and the associated economic

implications. The quality of information available

for system modelling is often not good enough to

be presented as either deterministic numbers or

probability distributions. Instead, some uncer-tainties can only be quantified as intervals or vague

values. The IFTSP can handle various uncertain-

ties described as probability and possibility distri-

butions, as well as discrete intervals.

The IFTSP can directly incorporate uncertain-

ties within its optimization framework. Its solu-

tions are presented by combinations of

deterministic, interval and distribution informa-tion, and thus offer flexibility in result interpreta-

tion and decision-alternative generation. Outputs

of the IFTSP model can reflect fluctuations in

system benefit (or cost) due to implementing dif-

ferent water-management policies; moreover, the

IFTSP solutions contain information of system-

failure risk under varying water-management

conditions. At the same time, the IFTSP outputsare unique since they are based on submodels (32)

and (33) as presented in Section 2.2; these two

submodels are deterministic linear programming

problems, with each of them having a unique,

global optimum. Thus, its solutions can provide

bases for selecting desired water-management

policies and plans with reasonable benefit and risk

levels.

5. Conclusions

An interval fuzzy two-stage stochastic pro-

gramming (IFTSP) model has been developed for

water resource management under uncertainty.

The model improves upon the existing fuzzy,interval and two-stage programming approaches

by allowing uncertainties presented as both fuzzy/

random distributions and discrete intervals to be

effectively incorporated within the optimization

framework. A new concept of interval fuzzy

membership function is introduced to the IFTSP


modelling process to reflect the complexities of thesystem uncertainties. In its solution process, the

IFTSP model is transformed into two determinis-

tic submodels, which correspond to the lower and

upper bounds of the desired objective. This

transformation is based on an interactive algo-

rithm. Interval solutions, which are stable in the

given decision space with an associated level of

system-failure risk, can then be obtained by solv-ing the two submodels sequentially.

Compared with the conventional two-stage

programming method, the IFTSP has an advan-

tage in reflecting the complexities of system

uncertainties presented as fuzzy, stochastic and

interval parameters as well as their combinations.

The IFTSP solutions provide not only decision-

variable solutions presented as stable intervals butalso the associated risk levels in violating the

objective aspiration and constraints as defined by

k�opt. Thus, the IFTSP approach provides more

information regarding trade-offs among system

benefits, certainty and reliability. As the actual

value of each variable or parameter varies within

its two bounds, the system benefit may change

correspondingly between f �opt and f þ

opt with a vari-ety of reliability levels.

Although this study is the first attempt for

planning a water-resources-management system

through the IFTSP approach, the results suggest

that this hybrid technique is applicable and can

be extended to other problems that involve poli-

cies with multi-criteria and multi-stage concerns,

as well as uncertainties presented in different for-mats.

Acknowledgements

The authors would like to thank the anony-

mous reviewers for their insightful and helpful

comments and suggestions that were very helpfulfor improving the manuscript. Thanks are also due

to the National High Technology Research

Foundation (No. 2001AA644020), the Natural

Science Foundation (No. 50225926) and the Nat-

ural Science and Engineering Research Council of

Canada.

References

Abrishamchi, A., Marino, M.A., Afshar, A., 1991. Reservoir

planning for irrigation district. Journal of Water Resources

Planning and Management 117, 74–85.

Anderson, L.E., 1968. A mathematical model for the optimi-

zation of a waste management system. SERL Report No.

68-1, Sanitary Engineering Research Laboratory, University

of California, Berkeley, CA, USA.

Babaeyan-Koopaei, K., Ervine, D.A., Pender, G., 2003. Field

measurements and flow modeling of overbank flows in

River Severn, UK. Journal of Environmental Informatics 1,

28–36.

Barbosa, P.S.F., Pricilla, R.P., 2002. A linear programming

model for cash flow management in the Brazilian construc-

tion industry. Journal of Planning Literature 16, 339–492.

Beraldi, P., Grandinetti, L., Musmanno, R., Triki, C., 2000.

Parallel algorithms to solve two-stage stochastic linear

programs with robustness constrains. Parallel Computing

26, 1889–1908.

Birge, J.R., 1985. Decomposition and partitioning methods for

multistage stochastic linear programs. Operations Research

33, 989–996.

Birge, J.R., Louveaux, F.V., 1988. A multicut algorithm for

two-stage stochastic linear programs. European Journal of

Operational Research 34, 384–392.

Byun, D.W., Kim, S.T., Cheng, F.Y., Kim, S.B., Cuclis, A.,

Moon, N.K., 2003. Information infrastructure for air

quality modeling and analysis: Application to the Hous-

ton–Galveston ozone non-attainment area. Journal of

Environmental Informatics 2, 38–57.

Cai, X., McKinney, D.C., Lasdon, L.S., 2001. Solving nonlin-

ear water management models using a combined genetic

algorithm and linear programming approach. Advances in

Water Resources 24 (6), 667–676.

Chang, N.B., Chen, W.C., 2000. Fuzzy controller design for

municipal incinerators with the aid of genetic algorithms

and genetic programming techniques. Waste Management

& Research ISWA 18, 429–443.

Chang, N.B., Wen, C.G., Chen, Y.L., Yong, Y.C., 1996. A grey

fuzzy multiobjective programming approach for the optimal

planning of a reservoir watershed, part A: Theoretical

development. Water Research 30, 2329–2340.

Dai, L., Chen, C.H., Birge, J.R., 2000. Convergence properties

of two-stage stochastic programming. Journal of Optimiza-

tion Theory and Applications 106, 489–509.

Darby-Dowman, K., Barker, S., Audsley, E., Parsons, D.,

2000. A two-stage stochastic programming with recourse

model for determining robust planting plans in horticul-

ture. Journal of the Operational Research Society 51, 83–

95.

Edirisinghe, N.C.P., Ziemba, W.T., 1994. Bounds for two-stage

stochastic programs with fixed resources. Mathematics of

Operation Research 19, 292–313.

Eiger, G., Shamir, U., 1991. Optimal operation of reservoirs by

stochastic programming. Engineering Optimization 17, 293–

312.


Ferrero, R.W., Rivera, J.F., Shahidehpour, S.M., 1998. A

dynamic programming two-stage algorithm for long-term

hydrothermal scheduling of multireservoir systems. IEEE

Transactions on Power Systems 13, 1534–1540.

Gang, D.C., Clevenger, T.E., Banerji, S.K., 2003. Modeling

chlorine decay in surface water. Journal of Environmental

Informatics 1, 21–27.

Gassmann, H.I., 1990. MSLIP: A computer code for the

multistage stochastic linear programming problem. Math-

ematical Programming 47, 407–423.

Guo, H.C., Liu, L., Huang, G.H., Zao, R., Yin, Y.Y., 2001. A

system dynamics approach for regional environmental

planning and management: A study for the Lake Erhai

Basin. Journal of Environmental Management 61, 93–112.

Haurie, A., Moresino, F., 2000. A stochastic programming

approach to manufacturing flow control. IIE Transactions

32, 907–920.

Hof, J., Bevers, M., 2000. Forestry––Direct spatial optimization

in natural resource management: Four linear programming

examples. Annals of Operations Research 95, 67–82.

Howe, B., Maier, D., Baptista, A., 2003. A language for spatial

data manipulation. Journal of Environmental Informatics 2,

23–37.

Huang, G.H., 1996. IPWM: An interval-parameter water

quality management model. Engineering Optimization 26,

79–103.

Huang, G.H., Chang, N.B., 2003. The perspectives of environ-

mental informatics and systems analysis. Journal of Envi-

ronmental Informatics 1, 1–6.

Huang, G.H., Loucks, D.P., 2000. An inexact two-stage

stochastic programming model for water resources man-

agement under uncertainty. Civil Engineering and Environ-

mental Systems 17, 95–118.

Huang, G., Moore, R.D., 1993. Grey linear programming, its

solving approach, and its application. International Journal

of Systems Science 24, 159–172.

Huang, G., Baetz, B.W., Patry, G.G., 1993. A grey fuzzy linear

programming approach for municipal solid waste manage-

ment planning under uncertainty. Civil Engineering Systems

10, 123–146.

Huang, G., Baetz, B.W., Patry, G.G., 1994. Capacity planning

for municipal solid waste management systems under

uncertainty––A grey fuzzy dynamic programming (GFDP)

approach. Journal of Urban Planning and Development

120, 132–156.

Huang, G., Baetz, B.W., Patry, G.G., 1995. Grey integer

programming: An application to waste management plan-

ning under uncertainty. European Journal of Operational

Research 83, 594–620.

Jairaj, P.G., Vedula, S., 2000. Multireservoir system optimiza-

tion using fuzzy mathematical programming. Water Re-

sources Management 14, 457–472.

Kall, P., 1979. Computational methods for solving two-stage

stochastic linear programming problems. ZAMP 30, 261–

271.

Kira, D., Kusy, M., Rakita, I., 1997. A stochastic linear

programming approach to hierarchical production plan-

ning. The Journal of the Operational Research Society 48,

207–211.

Kulcar, T., 1996. Optimizing solid waste collection in Brussels:

Theory and methodology. European Journal of Operational

Research 90, 71–77.

Liu, L., Hao, R.X., Cheng, S.Y., 2003. A possibilistic analysis

approach for assessing environmental risks from drinking

groundwater at petroleum-contaminated sites. Journal of

Environmental Informatics 2, 31–37.

Louveaux, F.V., 1980. A solution method for multistage

stochastic programs with recourse with application to an

energy investment problem. Operations Research 28, 889–

897.

Luo, B., Maqsood, I., Yin, Y.Y., Huang, G.H., Cohen, S.J.,

2003. Adaption to climate change through water trading

under uncertainty––An inexact two-stage nonlinear pro-

gramming approach. Journal of Environmental Informatics

2, 58–68.

Lustig, I.J., Mulvey, J.M., Carpenter, T.J., 1991. Formulating

two-stage stochastic programs for interior point methods.

Operations Research 39, 757–763.

Needham, J.T., Watkins Jr., D.W., Lund, J.R., Nanda, S.K.,

2000. Linear programming for flood control in the Iowa and

Des Moines Rivers. Journal of Water Resources Planning

and Management 126, 118–127.

Pereira, M.V.F., Pinto, L.M.V.G., 1991. Multi-stage stochastic

optimization applied to energy planning. Mathematical

Programming 52, 359–375.

Psilovikos, A.A., 1999. Optimization models in groundwater

management, based on linear and mixed integer program-

ming: An application to a Greek hydrogeological basin.

Physics and Chemistry of the Earth 24, 139–144.

Russell, S.O., Campbell, P.F., 1996. Reservoir operating rules

with fuzzy programming. Journal of Water Resources

Planning and Management 122, 165–170.

Ruszczynski, A., Swietanowski, A., 1997. Accelerating the

regularized decomposition method for two-stage stochastic

linear problems. European Journal of Operational Research

101, 328–342.

Sen, S., 1993. Subgradient decomposition and differentiability

of the recourse function of a two stage stochastic linear

program. Operations Research Letters 13, 143–148.

Shih, L.-H., 1999. Cement transportation planning via fuzzy

linear programming. International Journal of Production

Economics 58 (3), 277–289.

Wagner, J.M., Shamir, U., Marks, D.H., 1994. Containing

groundwater contamination: Planning models using sto-

chastic programming with recourse. European Journal of

Operational Research 7, 1–26.

Wang, D., Adams, B.J., 1986. Optimization of real-time

reservoir operations with Markov decision processes. Water

Resources Research 22, 345–352.

Wang, X.H., Du, C.M., 2003. An internet based flood warning

system. Journal of Environmental Informatics 2, 48–56.

Wang, L.Z., Fang, L., Hipel, K.W., 2003. Water resources

allocation: A cooperative game theoretic approach. Journal

of Environmental Informatics 2, 11–22.


Yeomans, J.S., Huang, G.H., 2003. An evolutionary grey, hop,

skip, and jump approach: Generating alternative policies for

the expansion of waste management. Journal of Environ-

mental Informatics 1, 37–51.

Yeomans, J.S., Huang, G.H., Yoogalingam, R., 2003. Com-

bining simulation with evolutionary algorithms for optimal

planning under uncertainty: An application to municipal

solid waste management planning in the regional munici-

pality of Hamilton–Wentworth. Journal of Environmental

Informatics 2, 11–30.

Yoshitomi, Y., Ikenoue, H., Takeba, T., Tomita, S., 2000.

Genetic algorithm in uncertain environments for solving

stochastic programming problem. Journal of the Operations

Research Society of Japan 43, 266–290.

Zhao, G., 2001. A log-barrier method with benders decompo-

sition for solving two-stage stochastic linear programs.

Mathematical Programming 90, 507–536.

Zimmermann, H., 1985. Fuzzy Set Theory and its Applications.

Kluwer–Nijhoff Publishing.

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